Find the differential equation by eliminating the arbitrary constant and satisfied by the equation $y={\tan }^{-1}x+c{e}^{-{\tan }^{-1}x}$, $c$ is arbitrary constant.

Find the differential equation by eliminating the arbitrary constant and satisfied by the equation $x^2+{\left(y-b\right)}^2=1$, $b$ is arbitrary constant.

Find the differential equation by eliminating the arbitrary constant and satisfied by the following equation ${\left(x-a\right)}^2-y^2=1$, $a$ is arbitrary constant.

Find the differential equation by eliminating the arbitrary constant and satisfied by the equation $x^2+y^2=a{x}^3$, $a$ is arbitrary constant.

Find the differential equation by eliminating the arbitrary constant and satisfied by the equation $\frac{x}{a}+\frac{y}{b}=1$, $a \& b$ are arbitrary constants.

Find the differential equation by eliminating the arbitrary constants and satisfied by the equation $y^2=4a\left(x-b\right)$, $a \& b$ are arbitrary constants.

Find the differential equation by eliminating the arbitrary constants and satisfied by the equation ${\left(y-b\right)}^2=4\left(x-a\right)$, $a \& b$ are arbitrary constants.

Find the differential equation by eliminating the arbitrary constants and satisfied by the equation $y^2=a\left(b^2-x^2\right)$, $a \& b$ are arbitrary constants.

Find the differential equation by eliminating the arbitrary constants and satisfied by the equation $y=a\cos \left(x+b\right)$, $a \& b$ are arbitrary constants.